A ball bounces back up $\frac{2}{3}$ of the height from which it falls. If the ball is dropped from a height of $243$ cm, after how many bounces does the ball first rise less than $30$ cm?
Solution: If you call the number of bounces $b$, then this problem can be phrased as: what is the minimum $b$, such that $243\cdot \left(\frac{2}{3}\right)^b < 30 \rightarrow \left(\frac{2}{3}\right)^b < \frac{30}{243}$, at which point you can take the logarithm of both sides and solve for it exactly, but given that you can use your calculator, it's much easier to just multiply $\frac{2}{3}$ with itself repeatedly until the product is less than $\frac{30}{243}$.  Regardless of how you do it, you will get $b = \boxed{6}$.